Symmetric hypergraph theorem

The Symmetric hypergraph theorem is a theorem in combinatorics that puts an upper bound on the chromatic number of a graph (or hypergraph in general). The original reference for this paper is unknown at the moment, and has been called folklore.^[1]

Statement

A group ${\displaystyle G}$ acting on a set ${\displaystyle S}$ is called transitive if given any two elements ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle S}$, there exists an element ${\displaystyle f}$ of ${\displaystyle G}$ such that ${\displaystyle f(x)=y}$. A graph (or hypergraph) is called symmetric if its automorphism group is transitive.

Theorem. Let ${\displaystyle H=(S,E)}$ be a symmetric hypergraph. Let ${\displaystyle m=|S|}$, and let ${\displaystyle \chi (H)}$ denote the chromatic number of ${\displaystyle H}$, and let ${\displaystyle \alpha (H)}$ denote the independence number of ${\displaystyle H}$. Then

$${\displaystyle \chi (H)\leq 1+{\frac {\ln {m}}{-\ln {(1-\alpha (H)/m)}}}}$$

Applications

This theorem has applications to Ramsey theory, specifically graph Ramsey theory. Using this theorem, a relationship between the graph Ramsey numbers and the extremal numbers can be shown (see Graham-Rothschild-Spencer for the details).

See also

• Ramsey theory

Notes

1. R. Graham, B. Rothschild, J. Spencer. Ramsey Theory. 2nd ed., Wiley, New-York, 1990.